Subgroups — #14279929

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₅	C₂	1	2	yes	maximal
3	C₂×#91115	C₂	1	4	yes	maximal
4	#10602328	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂³×Dic₁₅	C₂²	1	4	yes	—
8	C₂×C₃×C₄²×C₅	—	2	4	no	—
9	C₂×C₃×C₄²×C₅	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₅	—	2	4	no	—
11	C₂×C₄⋊Dic₁₅	—	2	4	no	—
12	C₂×C₄⋊Dic₁₅	C₄	1	8	yes	complemented
13	#91115	—	2	8	no	—
14	#71535	—	5	1	no	maximal, Hall
15	C₂²×C₄²×C₅	D₃	1	1	yes	characteristic
16	C₂²×C₄⋊Dic₅	—	3	2	no	—
17	C₂×#3963	—	3	4	no	—
18	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	2	4	no	—
20	C₂²×C₃×C₄×C₅	—	2	4	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	4	yes	—
28	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
29	C₂²×Dic₁₅	—	2	4	no	—
30	C₂²×Dic₁₅	—	2	4	no	—
31	C₂²×Dic₁₅	—	2	8	no	—
32	C₂²×Dic₁₅	—	2	8	no	—
33	C₂²×Dic₁₅	C₂×C₄	1	8	yes	—
34	C₃×C₄²×C₅	—	2	8	no	—
35	C₄⋊Dic₁₅	—	2	16	no	—
36	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic
37	C₂²×C₄⋊Dic₃	—	5	2	no	—
38	C₂×C₃⋊#95	—	5	4	no	—
39	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂³×Dic₅	—	3	4	no	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	Dic₃	1	4	yes	complemented
44	C₂×C₄⋊Dic₅	—	3	8	no	—
45	C₂×C₄⋊Dic₅	—	6	4	no	—
46	C₂×C₄⋊Dic₅	—	6	4	no	—
47	#3963	—	6	8	no	—
48	C₄⋊#95	—	15	1	no	Sylow
49	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
51	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
56	C₂³×C₃×C₅	#41	1	4	yes	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	8	no	—
60	C₂×C₃×C₄×C₅	—	2	8	no	—
61	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	8	yes	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	8	yes	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	8	yes	complemented
67	C₂×Dic₁₅	—	2	8	no	—
68	C₂×Dic₁₅	—	2	16	no	—
69	C₂×Dic₁₅	—	4	8	no	—
70	C₂×Dic₁₅	—	4	4	no	—
71	C₂×Dic₁₅	—	4	4	no	—
72	C₂³×C₃×C₄	C₂×D₅	1	2	yes	—
73	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
74	C₂³×Dic₃	—	5	4	no	—
75	C₂×C₃×C₄²	—	2	4	no	—
76	C₂×C₃×C₄²	Dic₅	1	4	yes	complemented
77	C₂×C₄⋊Dic₃	—	5	8	no	—
78	C₂×C₄⋊Dic₃	—	10	4	no	—
79	C₂×C₄⋊Dic₃	—	10	4	no	—
80	C₃⋊#95	—	10	8	no	—
81	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	Dic₆	1	4	yes	—
85	C₂²×C₄×C₅	Dic₆	1	2	yes	—
86	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
87	C₂²×C₄×C₅	D₁₂	1	4	yes	—
88	C₂²×C₄×C₅	D₁₂	1	2	yes	—
89	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
90	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
91	C₂²×C₄×C₅	#67	1	4	yes	—
92	C₂²×Dic₅	—	3	8	no	—
93	C₂²×Dic₅	—	6	8	no	—
94	C₂²×Dic₅	—	6	4	no	—
95	C₂²×Dic₅	—	6	8	no	—
96	C₂²×Dic₅	—	6	4	no	—
97	C₄²×C₅	—	2	8	no	—
98	C₄⋊Dic₅	—	6	16	no	—
99	C₂²×C₄⋊C₄	—	15	2	no	—
100	C₂²×C₄²	D₁₅	1	1	yes	characteristic
101	C₂×#95	—	15	4	no	—
102	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
108	C₂²×C₃×C₅	#117	1	2	yes	—
109	C₂²×C₃×C₅	C₄×D₄	1	4	yes	—
110	C₂²×C₃×C₅	C₄×Q₈	1	4	yes	—
111	C₂²×C₃×C₅	#121	1	4	yes	—
112	C₂²×C₃×C₅	C₂²⋊Q₈	1	4	yes	—
113	C₂²×C₃×C₅	#124	1	2	yes	—
114	C₂²×C₃×C₅	#125	1	2	yes	—
115	C₂²×C₃×C₅	#127	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
117	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
118	C₃×C₄×C₅	—	2	8	no	—
119	C₃×C₄×C₅	#95	1	8	yes	complemented
120	Dic₁₅	—	4	8	no	—
121	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	4	no	—
123	C₂²×C₃×C₄	—	2	4	no	—
124	C₂²×C₃×C₄	Dic₁₀	1	4	yes	—
125	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
126	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
127	C₂²×C₃×C₄	D₂₀	1	4	yes	—
128	C₂²×C₃×C₄	D₂₀	1	2	yes	—
129	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
130	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
131	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
132	C₂²×Dic₃	—	5	8	no	—
133	C₂²×Dic₃	—	10	8	no	—
134	C₂²×Dic₃	—	10	4	no	—
135	C₂²×Dic₃	—	10	8	no	—
136	C₂²×Dic₃	—	10	4	no	—
137	C₃×C₄²	—	2	8	no	—
138	C₄⋊Dic₃	—	10	16	no	—
139	C₂³×C₅	C₂×Dic₆	1	2	yes	—
140	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
141	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
142	C₂³×C₅	C₂×D₁₂	1	2	yes	—
143	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
144	C₂³×C₅	#235	1	4	yes	—
145	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
146	C₂³×C₅	C₂×#67	1	2	yes	—
147	C₂×C₄×C₅	—	2	8	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	8	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
152	C₂×C₄×C₅	Dic₃⋊C₄	1	8	yes	—
153	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
154	C₂×C₄×C₅	C₄⋊Dic₃	1	8	yes	complemented
155	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	8	yes	—
156	C₂×C₄×C₅	#217	1	4	yes	—
157	C₂×Dic₅	—	6	16	no	—
158	C₂×Dic₅	—	6	8	no	—
159	C₂×Dic₅	—	12	4	no	—
160	C₂×Dic₅	—	12	4	no	—
161	C₂×Dic₅	—	12	8	no	—
162	C₂³×C₄	—	15	4	no	—
163	C₂³×C₄	C₂×D₁₅	1	2	yes	—
164	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	15	8	no	—
166	C₂×C₄⋊C₄	—	30	4	no	—
167	C₂×C₄⋊C₄	—	30	4	no	—
168	C₂×C₄²	—	2	4	no	—
169	C₂×C₄²	Dic₁₅	1	4	yes	complemented
170	#95	—	30	8	no	—
171	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
172	C₂×C₃×C₅	C₄×C₄⋊C₄	1	2	yes	—
173	C₂×C₃×C₅	#380	1	2	yes	—
174	C₂×C₃×C₅	#382	1	1	yes	characteristic
175	C₂×C₃×C₅	#384	1	1	yes	characteristic
176	C₂×C₃×C₅	#389	1	4	yes	—
177	C₂×C₃×C₅	#390	1	2	yes	—
178	C₂×C₃×C₅	#391	1	2	yes	—
179	C₂³×C₃	C₂×Dic₁₀	1	2	yes	—
180	C₂³×C₃	C₂×Dic₁₀	1	1	yes	characteristic
181	C₂³×C₃	C₂×C₄×D₅	1	2	yes	—
182	C₂³×C₃	C₂×D₂₀	1	2	yes	—
183	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
184	C₂³×C₃	#709	1	4	yes	—
185	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
186	C₂³×C₃	C₂×C₂²⋊D₅	1	2	yes	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
192	C₂×C₃×C₄	#683	1	8	yes	—
193	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
194	C₂×C₃×C₄	C₄⋊Dic₅	1	8	yes	complemented
195	C₂×C₃×C₄	#685	1	8	yes	—
196	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
197	C₂×Dic₃	—	10	16	no	—
198	C₂×Dic₃	—	10	8	no	—
199	C₂×Dic₃	—	20	4	no	—
200	C₂×Dic₃	—	20	4	no	—
201	C₂×Dic₃	—	20	8	no	—
202	C₂²×C₅	C₄×Dic₆	1	4	yes	—
203	C₂²×C₅	#869	1	2	yes	—
204	C₂²×C₅	#869	1	1	yes	characteristic
205	C₂²×C₅	#870	1	2	yes	—
206	C₂²×C₅	C₄×D₁₂	1	4	yes	—
207	C₂²×C₅	#874	1	1	yes	characteristic
208	C₂²×C₅	#875	1	2	yes	—
209	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
210	C₂²×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
211	C₂²×C₅	#924	1	4	yes	—
212	C₂²×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
213	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂²×C₅	#926	1	2	yes	—
215	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
216	C₂²×C₅	#930	1	4	yes	—
217	C₂²×C₅	C₂×#217	1	1	yes	characteristic
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	C₃⋊#95	1	8	yes	complemented
220	Dic₅	—	12	8	no	—
221	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	15	8	no	—
225	C₂²×C₄	—	30	8	no	—
226	C₂²×C₄	—	30	4	no	—
227	C₂²×C₄	—	30	8	no	—
228	C₂²×C₄	—	30	4	no	—
229	C₂²×C₄	Dic₃₀	1	4	yes	—
230	C₂²×C₄	Dic₃₀	1	2	yes	—
231	C₂²×C₄	C₄×D₁₅	1	4	yes	—
232	C₂²×C₄	D₆₀	1	4	yes	—
233	C₂²×C₄	D₆₀	1	2	yes	—
234	C₂²×C₄	C₂×Dic₁₅	1	4	yes	—
235	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
236	C₂²×C₄	#1220	1	4	yes	—
237	C₄⋊C₄	—	30	16	no	—
238	C₄²	—	2	8	no	—
239	C₃×C₅	C₄⋊#95	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	C₄×Dic₁₀	1	4	yes	—
241	C₂²×C₃	#4015	1	2	yes	—
242	C₂²×C₃	#4015	1	1	yes	characteristic
243	C₂²×C₃	#4016	1	2	yes	—
244	C₂²×C₃	C₄×D₂₀	1	4	yes	—
245	C₂²×C₃	#4020	1	1	yes	characteristic
246	C₂²×C₃	#4021	1	2	yes	—
247	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
248	C₂²×C₃	C₂×#683	1	2	yes	—
249	C₂²×C₃	#4070	1	4	yes	—
250	C₂²×C₃	C₂×C₄⋊Dic₅	1	2	yes	—
251	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
252	C₂²×C₃	#4072	1	2	yes	—
253	C₂²×C₃	C₂×#685	1	2	yes	—
254	C₂²×C₃	#4076	1	4	yes	—
255	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
256	C₃×C₄	—	2	8	no	—
257	C₃×C₄	#3963	1	8	yes	complemented
258	Dic₃	—	20	8	no	—
259	C₂×C₅	#4928	1	4	yes	—
260	C₂×C₅	#4929	1	2	yes	—
261	C₂×C₅	C₄×C₄⋊Dic₃	1	2	yes	—
262	C₂×C₅	#4935	1	1	yes	characteristic
263	C₂×C₅	#4936	1	1	yes	characteristic
264	C₂×C₅	#4939	1	2	yes	—
265	C₂×C₅	C₂×C₃⋊#95	1	1	yes	characteristic
266	C₂×C₅	#5212	1	2	yes	—
267	C₂³	C₂×Dic₃₀	1	2	yes	—
268	C₂³	C₂×Dic₃₀	1	1	yes	characteristic
269	C₂³	C₂×C₄×D₁₅	1	2	yes	—
270	C₂³	C₂×D₆₀	1	2	yes	—
271	C₂³	C₂×D₆₀	1	1	yes	characteristic
272	C₂³	#6817	1	4	yes	—
273	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
274	C₂³	C₂×#1220	1	2	yes	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	30	16	no	—
280	C₂×C₄	—	30	8	no	—
281	C₂×C₄	—	60	4	no	—
282	C₂×C₄	—	60	4	no	—
283	C₂×C₄	—	60	8	no	—
284	C₂×C₄	C₄×Dic₁₅	1	4	yes	—
285	C₂×C₄	#6712	1	8	yes	—
286	C₂×C₄	C₄⋊Dic₁₅	1	4	yes	—
287	C₂×C₄	C₄⋊Dic₁₅	1	8	yes	complemented
288	C₂×C₄	#6714	1	8	yes	—
289	C₂×C₄	#6719	1	4	yes	—
290	C₂×C₃	#65217	1	4	yes	—
291	C₂×C₃	#65218	1	2	yes	—
292	C₂×C₃	C₄×C₄⋊Dic₅	1	2	yes	—
293	C₂×C₃	#65224	1	1	yes	characteristic
294	C₂×C₃	#65225	1	1	yes	characteristic
295	C₂×C₃	#65228	1	2	yes	—
296	C₂×C₃	C₂×#3963	1	1	yes	characteristic
297	C₂×C₃	#65501	1	2	yes	—
298	C₅	#71535	1	1	yes	characteristic, complemented, Sylow
299	C₂²	C₄×Dic₃₀	1	4	yes	—
300	C₂²	#91758	1	2	yes	—
301	C₂²	#91758	1	1	yes	characteristic
302	C₂²	#91759	1	2	yes	—
303	C₂²	C₄×D₆₀	1	4	yes	—
304	C₂²	#91763	1	1	yes	characteristic
305	C₂²	#91764	1	2	yes	—
306	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
307	C₂²	C₂×#6712	1	2	yes	—
308	C₂²	#91813	1	4	yes	—
309	C₂²	C₂×C₄⋊Dic₁₅	1	2	yes	—
310	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
311	C₂²	#91815	1	2	yes	—
312	C₂²	C₂×#6714	1	2	yes	—
313	C₂²	#91819	1	4	yes	—
314	C₂²	C₂×#6719	1	1	yes	characteristic
315	C₄	—	2	8	no	—
316	C₄	—	60	8	no	—
317	C₄	#91115	1	8	yes	complemented
318	C₃	#10602328	1	1	yes	characteristic, complemented, Sylow
319	C₂	#11751029	1	4	yes	—
320	C₂	#11751030	1	2	yes	—
321	C₂	C₄×C₄⋊Dic₁₅	1	2	yes	—
322	C₂	#11751036	1	1	yes	characteristic
323	C₂	#11751037	1	1	yes	characteristic
324	C₂	#11751040	1	2	yes	—
325	C₂	C₂×#91115	1	1	yes	characteristic
326	C₂	#11751313	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle, prime radical, cosocle	C₂⁴×C₃×C₅	C₂³
hypercenter	C₂²×C₄²	D₁₅
hypocenter	C₃×C₅	C₄⋊#95
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. C₂×C₃×C₄²×C₅	C₄	semidirect
12. C₂×C₄⋊Dic₁₅	C₄	semidirect
43. C₂×C₄²×C₅	Dic₃	semidirect
66. C₂×C₃×C₄×C₅	C₄⋊C₄	semidirect
76. C₂×C₃×C₄²	Dic₅	semidirect
119. C₃×C₄×C₅	#95	semidirect
154. C₂×C₄×C₅	C₄⋊Dic₃	semidirect
169. C₂×C₄²	Dic₁₅	semidirect
194. C₂×C₃×C₄	C₄⋊Dic₅	semidirect
219. C₄×C₅	C₃⋊#95	semidirect
239. C₃×C₅	C₄⋊#95	semidirect
257. C₃×C₄	#3963	semidirect
287. C₂×C₄	C₄⋊Dic₁₅	semidirect
298. C₅	#71535	semidirect
317. C₄	#91115	semidirect
318. C₃	#10602328	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₄²×C₅	2
C₂²×C₄⋊Dic₁₅	2
C₂×#91115	2
#10602328	3
#71535	5